Deriving model-based T$_e$-consistent chemical abundances in ionised gaseous nebulae

The derivation of abundances in gaseous nebulae ionised by massive stars using optical collisionally excited emission lines is studied in this work comparing the direct or $T_e$ method with updated grids of photoionisation models covering a wide range of input conditions of O/H and N/O abundances and ionisation parameter. The abundances in a large sample of compiled objects with at least one auroral line are re-derived and later compared with the $\chi^2$ weighted-mean abundances from the models. The agreement between the abundances using the two methods both for O/H and N/O is excellent with no additional assumptions about the geometry or physics governing the HII regions. Although very inaccurate model-based O/H are obtained when no auroral lines are considered, this can be overcome assuming empirical laws between O/H, log $U$, and N/O to constrain the considered models. In this way, for 12+log(O/H) $>$ 8.0, a precision better than 0.1dex consistent with the direct method is attained. For very low-$Z$, models give higher O/H values and a high dispersion, possibly owing to the contamination of the low-excitation emission-lines. However, in this regime, the auroral lines are usually well-detected. The use of this procedure, in a publicly available script, HII-CHI-mistry}, leads to the derivation of abundances in faint/high redshift objects consistent with the direct method based on CELs.